Asymptotic spectral properties of totally symmetric multilevel Toeplitz matrices

Let n = (n1 , … , nk) be a multiindex and . We say that n → ∞ if ni → ∞, 1 i k. If r = (r1, … , rk) and s = (s1, … , sk), let r − s = ( r1 − s1 , … , s1 − sk ). We say that a multilevel Toeplitz matrix of the form is totally symmetric. Let Qk be the k-fold Cartesian product of Q = [−π, π] with itself, and let be the Fourier coefficients of a function f = f(θ1, … , θk) in L2(Qk) that is even in each variable θ1, … , θk, so that Tn is totally symmetric for every n. We associate the multiindex n with 2k multiindices m(n, p), 0 p 2k − 1, such that limn→∞κ(m (n, p))/κ(n) = 2−k, 0 p 2k − 1, and , and show that the singular values of Tn separate naturally into 2k sets with cardinalities κ(m(n, 0)), … , κ(m(n, 2k − 1)) such that the singular values in each set are associated with singular vectors exhibiting a particular type of symmetry. Our main result is that the singular values in and the singular values of Tm(n, p) are absolutely equally distributed with respect to the class of functions bounded and uniformly continuous on as n → ∞, 0 p 2k − 1. If f is real-valued, then an analogous result holds for the eigenvalues and eigenvectors of Tn.